A vortex sheet is a term used in
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
for a surface across which there is a
discontinuity in
fluid velocity
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
, such as in slippage of one layer of fluid over another. While the
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
ial components of the flow velocity are discontinuous across the vortex sheet, the
normal Normal(s) or The Normal(s) may refer to:
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* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
component of the flow velocity is continuous. The discontinuity in the tangential velocity means the flow has infinite
vorticity
In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wit ...
on a vortex sheet.
At high
Reynolds numbers
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domin ...
, vortex sheets tend to be unstable. In particular, they may exhibit
Kelvin–Helmholtz instability
The Kelvin–Helmholtz instability (after Lord Kelvin and Hermann von Helmholtz) is a fluid instability that occurs when there is velocity shear in a single continuous fluid or a velocity difference across the interface between two fluids. ...
.
The formulation of the vortex sheet equation of motion is given in terms of a complex coordinate
. The sheet is described parametrically by
where
is the arclength between coordinate
and a reference point, and
is time. Let
denote the strength of the sheet, that is, the jump in the tangential discontinuity. Then the velocity field induced by the sheet is
The integral in the above equation is a Cauchy principal value integral. We now define
as the integrated sheet strength or circulation between a point with arc length
and the reference material point
in the sheet.
As a consequence of Kelvin's circulation theorem, in the absence of external forces on the sheet, the circulation between any two material points in the sheet remains conserved, so
. The equation of motion of the sheet can be rewritten in terms of
and
by a change of variable. The parameter
is replaced by
. That is,
This nonlinear integro-differential equation is called the Birkoff-Rott equation. It describes the evolution of the vortex sheet given initial conditions. Greater details on vortex sheets can be found in the textbook by Saffman (1977).
Diffusion of a vortex sheet
Once a vortex sheet, it will diffuse due to viscous action. Consider a planar unidirectional flow at
,
:
impling the presence of a vortex sheet at
. The velocity discontinuity smooths out according to
[Drazin, P. G., & Riley, N. (2006). The Navier-Stokes equations: a classification of flows and exact solutions (No. 334). Cambridge University Press.]
:
where
is the
kinematic viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
. The only non-zero vorticity component is in the
direction, given by
:
.
Vortex sheet with periodic boundaries
A flat vortex sheet with periodic boundaries in the streamwise direction can be used to model a temporal free shear layer at high Reynolds number. Let us assume that the interval between the periodic boundaries is of length
. Then the equation of motion of the vortex sheet reduces to
Note that the integral in the above equation is a Cauchy principal value integral. The initial condition for a flat vortex sheet with constant strength is
. The flat vortex sheet is an equilibrium solution. However, it is unstable to infinitesimal periodic disturbances of the form
. Linear theory shows that the Fourier coefficient
grows exponentially at a rate proportional to
. That is, higher the wavenumber of a Fourier mode, the faster it grows. However, a linear theory cannot be extended much beyond the initial state. If nonlinear interactions are taken into account, asymptotic analysis suggests that for large
and finite